Video: Midpoint on the Coordinate Plane

In this video, we will learn how to find the coordinates of a midpoint between two points or those of an endpoint on the coordinate plane.

15:24

Video Transcript

In this video, we will learn how to find the coordinates of a midpoint between two points or those of an endpoint on the coordinate plane. Before we look at a coordinate plane, let’s look at a midpoint on a number line.

If we start with a number line and we have points at two and eight, where is the midpoint between two and eight? You might say, “Well, it’s going to be halfway between two and eight.” Someone else might say, “It’s the average of two and eight.” The average of two and eight is halfway between. To find the average of two and eight, we would add two plus eight and then divide by two, which equals 10 divided by two, which is five. And that means halfway between two and eight, the average of two and eight is five.

To find the midpoint on a coordinate plane, we use a really similar process. Let’s consider this line segment in a coordinate grid. If we label the blue point 𝐴, 𝐴 is located at two, two. And if we make the other endpoint 𝐵, 𝐵 is located at eight, six. And 𝐶 is the midpoint. This means that the distance from 𝐴 to 𝐶 is equal to the distance from 𝐶 to 𝐵. It also means that 𝐶 is halfway between 𝐴 and 𝐵 in the vertical direction and 𝐶 is halfway between 𝐴 and 𝐵 in the horizontal direction. And we write that mathematically like this.

To find the 𝑥-coordinate of the midpoint, we take the 𝑥-coordinates of the two endpoints and we average them. Remember that the endpoints are located at either end of the line segment. Then, the 𝑦-coordinate of the midpoint is found by taking the 𝑦-coordinates of the two endpoints and averaging those. In this case, we can let 𝐴 be 𝑥 one, 𝑦 one and 𝐵 be 𝑥 two, 𝑦 two. If we want to locate point 𝐶’s coordinates, we plug in the values we know from the endpoints. 𝑥 one is two. 𝑥 two is eight. 𝑦 one is two. 𝑦 two is six.

When we add those together, we get 10 over two, eight over two, which simplifies further to five, four. The midpoint is then located at the point five, four. Five is halfway between two and eight and four is halfway between two and six. Using this formula, we can find the midpoint if we’re given two coordinates. Or we can find the coordinates of an endpoint if we’re given one endpoint and the midpoint.

Let’s look at some examples.

On the graph, which point is halfway between one, eight and five, two?

We have the point one, eight — we could call it 𝐴 — and the point five, two — we can call 𝐵. And we want to know which point is halfway between the two points. The point halfway between these two points is called the midpoint. And to find the midpoint, we average the 𝑥-coordinates of the two endpoints. And then, we average the 𝑦-coordinates of the two endpoints. So, we’ll let point 𝐴 be 𝑥 one, 𝑦 one and point 𝐵 be 𝑥 two, 𝑦 two. Then to find the midpoint, we add one plus five and divide by two. And then, we add eight plus two and divide by two. This gives us six over two and 10 over two, which we reduce to three, five.

On the graph, the coordinate three, five is here. And if we look at this point, we see from 𝐵 that is up three and left two. And then, if we want to go from the midpoint to point 𝐴, again, we have up three and left two. This confirms that the distance from 𝐴 to the midpoint is the same as the distance from the midpoint to 𝐵. Halfway between one, eight and five, two is the point three, five.

Here’s another example. This time, we know the endpoint. But we’re missing part of the coordinates for the endpoints.

Consider the points 𝐴: 𝑥, seven; 𝐵: negative four, 𝑦; and 𝐶: two, five. Given that 𝐶 is the midpoint of line segment 𝐴𝐵, find the values of 𝑥 and 𝑦.

First, let’s list out what we know. We have line segment 𝐴𝐵 and 𝐶 is the midpoint point. Point 𝐴 is located at 𝑥, seven. Point 𝐵 is located at negative four, 𝑦. And point 𝐶 is located at two, five. At this point, you might be thinking, “Shouldn’t we try to graph these values?” But because we’re missing this 𝑥- and 𝑦-value, it’s not easy to graph this. So, let’s consider what we know about the midpoint.

The coordinates of the midpoint can be found by averaging the 𝑥-coordinates of the endpoints and the 𝑦-coordinates of the endpoints. And if the midpoint is two, five, then 𝑥 one plus 𝑥 two divided by two has to be equal to two. And 𝑦 one plus 𝑦 two divided by two has to be equal to five. So, we set up two separate equations, one that says 𝑥 one plus 𝑥 two over two equals two and one that says five equals 𝑦 one plus 𝑦 two over two. We’ll let 𝐴 be 𝑥 one, 𝑦 one and 𝐵 be 𝑥 two, 𝑦 two and then we plug in what we know.

Two is then equal to 𝑥 plus negative four divided by two and five is equal to seven plus 𝑦 divided by two. And now, we just need to solve for each variable. On the left, we multiply both sides of the equation by two, which will give us four equals 𝑥 plus negative four, which we can rewrite to say four equals 𝑥 minus four. Then add four to both sides. And we see that eight equals 𝑥 or, more commonly, 𝑥 equals eight. We follow the same procedure to solve for 𝑦. Multiply by two. Subtract seven from both sides. Three equals 𝑦. So, 𝑦 equals three. Since 𝐴 equals 𝑥 seven and 𝑥 equals eight, point 𝐴 is located at eight, seven. And since 𝐵 was located at negative four, 𝑦 and 𝑦 equals three, 𝐵 is located at negative four, three.

Here’s another example, where we already know the endpoint and we need to find the coordinates of the endpoints.

Find the point 𝐴 on the 𝑥-axis and the point 𝐵 on the 𝑦-axis such that three over two, negative five over two is the midpoint of line segment 𝐴𝐵.

First, let’s list out what we know. 𝐴 is located along the 𝑥-axis. 𝐵 is located along the 𝑦-axis. And we have a midpoint of three-halves, negative five-halves. At this point, we should think about what we know about the 𝑥- and 𝑦-axis. Values along the 𝑥-axis have a 𝑦-coordinate of zero. And that means if point 𝐴 is located along the 𝑥-axis, we don’t know where along the 𝑥-axis that is. But we do know it’s zero along the 𝑦-axis. We can write out the coordinates of point 𝐴 as 𝑎, zero. And in a similar way, points along the 𝑦-axis have zero as their 𝑥-coordinate. That means we can write point 𝐵 as being located at zero, 𝑏. And finally, we remember that we can find the midpoint by averaging the 𝑥-coordinates of the endpoints and the 𝑦-coordinates of the endpoints.

Using this information, we can set up two equations to solve for our missing values. We’ll let point 𝐴 be 𝑥 one, 𝑦 one and point 𝐵 be 𝑥 two, 𝑦 two. We plug that information into our midpoint formula. That means three-halves is equal to 𝑎 plus zero over two and negative five-halves equals zero plus 𝑏 over two. 𝑎 plus zero equals 𝑎. Three-halves must be equal to 𝑎 over two. And in the same way, negative five over two is equal to 𝑏 over two, which means 𝑎 equals three and 𝑏 equals negative five. We take this information and plug it back into our points. And we get the point three, zero and the point zero, negative five. These two points have the given midpoint.

Let’s look at a case where we can use what we know about midpoints to solve a problem.

A rectangular garden is next to a house along the road. In the garden is an orange tree seven meters from the house and three meters from the road. There is also an apple tree, five meters from the house and nine meters from the road. A fountain is placed halfway between the trees. How far is the fountain from the house and the road?

We have a garden that is next to a house along a road. In the garden, there are two trees, an orange tree that’s located seven meters from the house and three meters from the road. And then, there’s an apple tree, five meters from the house and nine meters from the road. A fountain is placed halfway between them. How can we find out how far the fountain is from the house and from the road?

We could write these distances in a coordinate form, where the 𝑥-coordinate is the distance from the house and the 𝑦-coordinate is the distance from the road. The orange tree is then located at seven, three. The apple tree is located at five, nine. And the fountain is the midpoint. And we know to find the midpoint, we add the 𝑥-coordinates of the endpoints and divide by two and then add the 𝑦-coordinates of the endpoints and divide by two. Which means the fountain is located at seven plus five divided by two and nine plus three divided by two, which is 12 over two, 12 over two or six, six. We know that our 𝑥-coordinate is the distance from the house and the 𝑦-coordinate is the distance from the road. Therefore, the fountain is located six meters from the house and six meters from the road.

In our last example, we’ll look at a midpoint that is the midpoint of two different line segments at the same time.

Suppose 𝐴 is negative seven, negative four; 𝐵 is six, negative nine; and 𝐷 is eight, negative two. If 𝐶 is the midpoint of both line segment 𝐴𝐵 and line segment 𝐷𝐸, find point 𝐸.

Let’s first sketch what we know. We have a line segment 𝐴𝐵 with the midpoint 𝐶. And 𝐶 is also the midpoint of line segment 𝐷𝐸. We’re given the coordinates for 𝐴, 𝐵, and 𝐷. Our end goal is to find the coordinates of point 𝐸. But before we can find 𝐸, we’ll need to know 𝐶. Once we find 𝐶, we can find 𝐸. And to do both of these things, we’ll need to remember that the midpoint formula looks like this.

The 𝑥-coordinate of the midpoint is found by taking the 𝑥-coordinates from the endpoints and dividing by two. And the 𝑦-coordinate of the midpoint is found by averaging the 𝑦-coordinates of the two endpoints. Since 𝐶 is the midpoint of 𝐴 and 𝐵, we’ll let 𝐴 be 𝑥 one, 𝑦 one and 𝐵 be 𝑥 two, 𝑦 two. The midpoint 𝐶 will be located at negative seven plus six over two, negative four plus negative nine over two. Negative seven plus six over two is negative one-half. And negative four plus negative nine is negative 13. So, the 𝑦-coordinate is negative 13 over two. Now, we know where 𝐶 is located. And we’re ready to think about 𝐸.

If 𝐶 is also the midpoint of 𝐷𝐸, then the coordinates of 𝐶 will be equal to the 𝑥-coordinates of 𝐷 and 𝐸 averaged together and the 𝑦-coordinates of 𝐷 and 𝐸 averaged together. We’re given the coordinates of 𝐷. That’s eight, negative two. And so, we plug that in. From here, we’ll make two separate equations. We’ll set negative one-half equal to eight plus the 𝑥-coordinate of 𝐸 over two. And negative 13 over two is equal to negative two plus the 𝑦-coordinate of 𝐸 over two. We’ll give ourselves a little bit more room.

Since all of the denominators are two, then the numerators are equal to each other. Negative one equals eight plus the 𝑥-coordinate of 𝐸. And to solve for that missing value, we subtract eight from both sides. And we see the 𝑥-coordinate for point 𝐸 is negative nine. To solve for the 𝑦-coordinate of point 𝐸, we add two to both sides. The 𝑦-coordinate of 𝐸 is negative 11. In coordinate form, point 𝐸 is located at negative nine, negative 11.

Each of these examples has shown us that when you have the endpoints 𝑥 one, 𝑦 one; 𝑥 two, 𝑦 two, the midpoint is found by adding 𝑥 one plus 𝑥 two and dividing by two then 𝑦 one plus 𝑦 two divided by two. We can also use this formula to find an endpoint if we know one endpoint and the midpoint. Using this formula, you’re ready to try some on your own.

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